Definition 2

Let I be a set; let M(I) (resp. Mo(I), resp. $\mathbf{N}^{(I)}$) denote the free magma (resp. free monoid, resp. free commutative monoid) derived from I. The algebra of M(I) (resp. Mo(I), resp. $\mathbf{N}^{(I)}$) over A is called the free algebra (resp. free associative algebra, resp. free commutative associative algebra (or, by an abuse of language, free commutative algebra)) of the set I over the ring A.

We shall denote the free algebra (resp. free associative algebra, resp. free commutative algebra) of I over A by Lib_A(I) (resp. Libas_A(I), resp. Libasc_A(I)). By composing the canonical mapping of I into M(I) (resp. Mo(I), resp. $\mathbf{N}^{(I)}$) with the canonical mapping of M(I) (resp. Mo(I), resp. $\mathbf{N}^{(I)}$) into Lib_A(I) (resp. Libas_A(I), resp. Libasc_A(I)), a canonical mapping is obtained of I into Lib_A(I) (resp. Libas_A(I), resp. Libasc_A(I)), which is injective if $A \neq \{0\}$. We shall denote the image of an element $i \in I$ under this canonical mapping by $X_i$ and we shall say that $X_i$ is the indeterminate of index i of Lib_A(I) (resp. Libas_A(I), resp. Libasc_A(I)).

As Mo(I) and $\mathbf{N}^{(I)}$ each have an identity element, Libas_A(I) and Libasc_A(I) are unital associative algebras and further Libasc_A(I) is commutative. If e is the unit element of Libas_A(I) (resp. Libasc_A(I)), the mapping $\alpha \mapsto \alpha e$ is an isomorphism of A onto a subring of the centre of Libas_A(I) (resp. Libasc_A(I)), which is identified with A (no. 1).